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I know I promised to talk about homotopy theory and n-categories, but I've gotten sidetracked into thinking about projective planes, so I'll talk about that this Week and go back to the other stuff later. Sorry, but if I don't talk about what intrigues me at the instant I'm writing this stuff, I can't get up the energy to write it.
So:
There are many kinds of geometry. After Euclidean geometry, one of the first to become popular was projective geometry. Projective geometry is the geometry of perspective. If you draw a picture on a piece of paper and view it from a slant, distances and angles in the picture will get messed up - but lines will still look like lines. This kind of transformation is called a "projective transformation". Projective geometry is the study of those aspects of geometry that are preserved by projective transformations.
Interestingly, 2-dimensional projective geometry has some curious features that don't show up in higher dimensions. To explain this, I need to tell you about projective planes.
I talked a bit about projective planes in "week106". The basic idea is to take the ordinary plane and add some points at infinity so that every pair of distinct lines intersects in exactly one point. Lines that were parallel in the ordinary plane will intersect at one of the points at infinity. This simplifies the axioms of projective geometry.
But what exactly do I mean by "the ordinary plane"? Well, ever since Descartes, most people think of the plane as R2, which consists of ordered pairs of real numbers. But algebraists also like to use C2, consisting of ordered pairs of complex numbers. For that matter, you could take any field F - like the rational numbers, or the integers modulo a prime - and use F2. Algebraic geometers call this sort of thing an "affine plane".
A projective plane is a bit bigger than an affine plane. For this, start with the 3-dimensional vector space F3. Then define the projective plane over F, denoted FP2, to be the space of lines through the origin in F3. You can show the projective plane is the same as the affine plane together with extra points, which play the role of "points at infinity".
In fact, you can generalize this a bit - you can make sense of the projective plane over F whenever F is a division ring! A division ring is a like a field, but where multiplication isn't necessarily commutative. The best example is the quaternions. In "week106" I talked about the real, complex and quaternionic projective planes, their symmetry groups, and their relation to quantum mechanics. Here's a good book about this stuff, emphasizing the physics applications:
1) V. S. Varadarajan, Geometry of Quantum Mechanics, Springer-Verlag, Berlin, 2nd ed., 1985.
So far, so good. But there's another approach to projective planes that's even more general. This approach goes back to Euclidean geometry: it's based on a list of axioms. In this approach, a projective plane consists of a set of "points", a set of "lines", and a relation which tells us whether or not a given point "lies on" a given line. I'm putting quotes around all these words, because in this approach they are undefined terms. All we get to work with are the following axioms:
A) Given two distinct points, there exists a unique line that both points lie on.
B) Given two distinct lines, there exists a unique point that lies on both lines.
C) There exist four points, no three of which lie on the same line.
D) There exist four lines, no three of which have the same point lying on them.
Actually we can leave out either axiom C) or axiom D) - the rest of the axioms will imply the one we leave out. It's a nice little exercise to convince yourself of this. I put in both axioms just to make it obvious that this definition of projective plane is "self-dual". In other words, if we switch the words "point" and "line" and switch who lies on who, the definition stays the same!
Duality is one of the great charms of the theory of projective planes: whenever you prove any theorem, you get another one free of charge with the roles of points and lines switched, thanks to duality. There are lots of different kinds of "duality" in mathematics, but this is probably the grand-daddy of them all.
Now, it's easy to prove that starting from any division ring F, we get a projective plane FP2 satisfying the above axioms. The fun part is to try to go the other way! Starting from a structure satisfying the above axioms, can you cook up a division ring that it comes from?
Well, starting from a projective plane, you can try to recover a division ring as follows. Pick a line and throw out one point - and call that point "the point at infinity". What's left is an "affine line" - let's call it L. Let's try to make L into a division ring. To do this, we first need to pick two different points in L, which we call 0 and 1. Then we need to cook up rules for adding and multiplying points on L.
For this, we use some tricks invented by the ancient Greeks!
This should not be surprising. After all, those dudes thought about arithmetic in very geometrical ways. How can you add points on a line using the geometry of the plane? Just ask any ancient Greek, and here's what they'll say:
First pick a line L' that's parallel to L - meaning that L and L' intersect only at the point at infinity. Then pick a line M that intersects L at the point 0 and L' at some point which we call 0'. We get a picture like this:
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----------0'------------------------------------------ L'
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M|
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----------0------------------------------------------ L
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Then, to add two points x and y on L, draw this picture:
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----------0'----------------|------------------------- L'
| \ | \
| \ | \
M| \N M'| \N'
| \ | \
----------0-----x-----------y-----z------------------ L
| \ | \
In other words, draw a line M' parallel to M through the point y,
draw a line N through x and 0', and draw a line N' parallel to N and
going through the point where M' and L' intersect. L and N' intersect
at the point called z... and we define this point to be x + y!
This is obviously the right thing, because the two triangles in the picture are congruent.
What about multiplication? Well, first draw a line L' that intersects our line L only at the point 0. Then draw a line M from the point 1 to some point 1' that's on L' but not on L:
L'/
/
/
1'
/|
/ |
/ |
----------0---1------------------------------------ L
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|M
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Then, to multiply x and y, draw this picture:
/
/|\
/ | \
/ | \
L'/ | \
/ | \
/ | \
1' | \
/|\ | \
/ | \N | \N'
/ | \ | \
----------0---1---x--y----------z------------------ L
| \ | \
|M |M'
| |
In other words, draw a line N though 1' and x, draw a line M' parallel
to M through the point y, and draw a line N' parallel to N through the
point where L' and M' intersect. L and N' intersect at the point called
z... and we define this point to be xy!
This is obviously the right thing, because the triangle containing the points 1 and x is similar to the triangle containing y and z.
So, now that we've cleverly figured out how to define addition and multiplication starting from a projective plane, we can ask: do we get a division ring?
And the answer is: not necessarily. It's only true if our projective plane is "Desarguesian". This is a special property named after an old theorem about the real projective plane, proved by Desargues. A projective plane is Desarguesian if Desargues' theorem holds for this plane.
But wait - there's an even more basic question we forgot to ask! Namely: was our ancient Greek method of defining addition and multiplication independent of the choices we made? We needed to pick some points and lines to get things going. If you think about it hard, these choices boil down to picking four points, no three of which lie on a line - exactly what axiom C) guarantees we can do.
Alas, it turns out that in general our recipe for addition and multiplication really depends on how we chose these four points. But if our projective plane is Desarguesian, it does not!
In fact, if we stick to Desarguesian projective planes, everything works very smoothly. For any division ring F the projective plane FP^2 is Desarguesian. Conversely, starting with a Desarguesian projective plane, we can use the ancient Greek method to cook up a division ring F. Best of all, these two constructions are inverse to each other - at least up to isomorphism.
At this point you should be pounding your desk and yelling "Great - but what does `Desarguesian' mean? I want the nitty-gritty details!"
Okay....
Given a projective plane, define a "triangle" to be three points that don't lie on the same line. Now suppose you have two triangles xyz and x'y'z'. The sides of each triangle determine three lines, say LMN and L'M'N'. If we're really lucky, the line through x and x', the line through y and y', and the line through z and z' will all intersect at the same point. We say that our projective plane is "Desarguesian" if whenever this happens, something else happens: the intersection of L and L', the intersection of M and M', and the intersection of N and N' all lie on the same line.
If you have trouble visualizing what I just said, take a look at this webpage, which also gives a proof of Desargues' theorem for the real projective plane:
2) Roger Mohr and Bill Trigs, Desargues' Theorem, http://spigot.anu.edu.au/people/samer/Research/Doc/ECV_Tut_Proj_Geom/node25.html
Desargues' theorem is a bit complicated, but one cool thing is that its converse is its dual. This is easy to see if you stare at it: "three lines intersecting at the same point" is dual to "three points lying on the the same line". Even cooler, Desargues' theorem implies its own converse! Thus the property of being Desarguesian is self-dual.
Another nice fact about Desarguesian planes concerns collineations. A "collineation" is a map from a projective plane to itself that preserves all lines. Collineations form a group, and this group acts on the set of all "quadrangles" - a quadrangle being a list of four points, no three of which lie on a line. Axiom C) says that every projective plane has at least one quadrangle. It turns out that if a projective plane is Desarguesian, the group of collineations acts transitively on the set of quadrangles: given any two quadrangles, there's a collineation carrying one to the other. This is the reason why the ancient Greek trick for adding and multiplying doesn't depend on the choice of quadrangle when our projective plane is Desarguesian!
An even more beautiful fact about Desarguesian planes concerns their relation to higher dimensions. Just as we defined projective planes through a list of axioms, we can also define projective spaces of any dimension n = 1,2,3,.... The simplest example is FPn - the space of lines through the origin in Fn+1, where F is some division ring. The neat part is that when n > 2, this is the only example. Moreover, any projective plane sittting inside one of these higher-dimensional projective spaces is automatically Desarguesian! So the non-Desarguesian projective planes are really freaks of dimension 2.
All this is very nice. But there are some obvious further questions, namely: what's special about projective planes that actually come from fields, and what can we say about non-Desarguesian projective planes?
The key to the first question is an old theorem proved by the last of the great Greek geometers, Pappus, in the 3rd century CE. It turns out that in any projective plane coming from a field, the Pappus Theorem holds. Conversely, any projective plane satisfying the Pappus Theorem comes from a unique field. We call such projective planes "Pappian".
The Pappus theorem will be too scary if I explain it using only words, so I'll tell you to look at a picture instead. The fun thing about this picture is that you can move the red and green points around with your mouse and see how things change:
3) Pappus' theorem (a JavaSketchPad demo by MathsNet), http://www.anglia.co.uk/education/mathsnet/dynamic/pappus.html
Now, what about the non-Desarguesian projective planes? If we try to get a division ring from an arbitrary projective plane, we fail miserably. However, we can still define addition and multiplication using the tricks described above. These operations depend crucially our choice of a quadrangle. But if we list all the axioms these operations satisfy, we get the definition of an algebraic gadget called a "ternary ring".
They're called "ternary rings" because they're usually described in terms of a ternary operation that generalizes xy + z. But the precise definition is too depressing for me give here. It's a classic example of what James Dolan calls "centipede mathematics", where you take a mathematical concept and see how many legs you can pull off before it can no longer walk. A ternary ring is like a division ring that can just barely limp along on its last legs.
I'm not a big fan of centipede mathematics, but there is one really nice example of a ternary ring that isn't a division ring. Namely, the octonions! These are almost a division ring, but their multiplication isn't associative.
I already talked about the octonions in "week59", "week61", "week104" and "week105". In "week106", I explained how you can define OP2, the projective plane over the octonions. This is the best example of a non-Desarguesian projective plane. One reason it's so great is that that its group of collineations is E6. E6 is one of the five "exceptional simple Lie groups" - mysterious and exciting things that deserve all the study they can get!
Next I want to talk about the relation between projective geometry and the other exceptional Lie groups, but first let me give you some references. To start, here's a great book on projective planes and all the curious centipede mathematics they inspire:
4) Frederick W. Stevenson, Projective Planes, W. H. Freeman and Company, San Francisco, 1972.
You'll learn all about nearfields, quasifields, Moufang loops, Cartesian groups, and so on.
Much of the same material is covered in these lectures by Hall, which are unfortunately a bit hard to find: 5) Marshall Hall, Projective Planes and Other Topics, California Institute of Technology, Pasadena, 1954.
For a more distilled introduction to the same stuff, try the last chapter of Hall's book on group theory:
6) Marshall Hall, The Theory of Groups, Macmillan, New York, 1959.
If you're only interested in Desarguesian projective planes, try this:
7) Robin Hartshorne, Foundations of Projective Geometry, Benjamin, New York, 1967.
In particular, this book gives a nice account of the collineation group in the Desarguesian case. The punchline is simple to state, so I'll tell you. Suppose F is a division ring. Then the collineation group of FP2 is generated by two obvious subgroups: PGL(3,F) and the automorphism group of F. The intersection of these two subgroups is the group of inner automorphisms of F.
If the above references are too intense, try this leisurely, literate introduction to the subject first:
8) Daniel Pedoe, An Introduction to Projective Geometry, Macmillan, New York, 1963.
And you're really interested in the finite projective planes, you can try this reference, which assumes very little knowledge of algebra:
9) A. Adrian Albert and Reuben Sandler, An Introduction to Finite Projective Planes, Holt, Rinehart and Winston, New York, 1968.
For a nice online introduction to projective geometry over the real numbers and its applications to image analysis, try this:
10) Roger Mohr and Bill Triggs, Projective geometry for image analysis, http://spigot.anu.edu.au/people/samer/Research/Doc/ECV_Tut_Proj_Geom/node1.html
Finally, for interesting relations between projective geometry and exceptional Lie groups, try this:
11) J. M. Landsberg and L. Manivel: The projective geometry of Freudenthal's magic square, preprint available as math.AG/9908039.
The Freudenthal-Tits magic square is a strange way of describing most of the exceptional Lie groups in terms of the real numbers, complex numbers, quaternions and octonions. In the usual way of describing it, you start with two of these division algebras, say F and F'. Then let J(F) be the space of 3x3 self-adjoint matrices with coefficients in F. This is a Jordan algebra with the product xy + yx. As mentioned in "week106", Jordan algebras have a lot to do with projective planes. In particular, the nontrivial projections in J(F) correspond to the 1- and 2-dimensional subspaces of F3, and thus to the points and lines in the projective plane PF2.
Next, let J_0(F) be the subspace of J(F) consisting of the traceless self-adjoint matrices. Also, let Im(F') be the space of pure imaginary element of K'. Finally, let the magic Lie algebra M(F,F') be given by
M(K,K') = Der(J(K)) + J_0(K) x Im(K') + Der(K')
Here + stands for direct sum, x stands for tensor product, and Der stands for the space of derivations of the algebra in question. It's actually sort of tricky to describe how to make M(F,F') into a Lie algebra, and I'm sort of tired, so I'll wimp out and tell you to read this stuff:
12) Hans Freudenthal, Lie groups in the foundations of geometry, Adv. Math. 1 (1964) 143.
13) Jacques Tits, Algebres alternatives, algebres de Jordan et algebres de Lie exceptionelles, Proc. Colloq. Utrecht, vol. 135, 1962.
14) R. D. Schafer, Introduction to Non-associative Algebras, Academic Press, 1966.
By the way, the paper by Freudenthal is a really mind-bending mix of Lie theory and axiomatic projective geometry, definitely worth looking at. Anyway, if you do things right you get the following square of Lie algebras M(F,F'):
F = R F = C F = H F = O
F' = R A1 A2 C3 F4
F' = C A2 A2+A2 A5 E6
F' = H C3 A5 B6 E7
F' = O F4 E6 E7 E8
Here R, C, H and O stand for the reals, complexes, quaternions and
octonions. If you don't know what all the Lie algebras in the square
are, check out "week64". (I
should admit that the above square is not very precise, because I don't
say which real forms of the Lie algebras in question are showing up.)
The first fun thing about this square is that F4, E6, E7 and E8 are four of the five exceptional simple Lie algebras - and the fifth one, G2, is just Der(O). So all the exceptional Lie algebras are related to the octonions! And the second fun thing about this square is that it's symmetrical along the diagonal, even though this is not at all obvious from the definition. This is what makes the square truly "magic".
I don't really understand the magic square, but it's on my to-do list. That's one reason I'm glad there's a new paper out that describes the magic square in a way that makes its symmetry manifest:
15) C. H. Barton and A. Sudbery, Magic squares of Lie algebras, preprint available as math.RA/0001083.
It also generalizes the magic square in a number of directions. But what I really want is for the connection between projective planes, division algebras, exceptional Lie groups and the magic square to becomes truly obvious to me. I'm a long way from that point.
Here's an interesting email from David Broadhurst about the failure of the Pappus theorem in quaternionic projective space:
Date: Fri, 3 Mar 2000 20:50:03 GMT From: D.Broadhurst@open.ac.uk (David Broadhurst) Subject: Paul Dirac and projective geometry John: Shortly before his death I spent a charming afternoon with Paul Dirac. Contrary to his reputation, he was most forthcoming. Among many things, I recall this: Dirac explained that while trained as an engineer and known as a physicist, his aesthetics were mathematical. He said (as I can best recall, nearly 20 years on): At a young age, I fell in love with projective geometry. I always wanted to use to use it in physics, but never found a place for it. Then someone told him that the difference between complex and quaternionic QM had been characterized as the failure of theorem in classical projective geometry. Dirac's face beamed a lovely smile: "Ah," he said, "it was just such a thing that I hoped to do". I was reminded of this when backtracking to your "week106", today. Best, David
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